I've never added cardboard to compost... I have lined my garden paths with it b4 covering with woodchips...
The only problem I ever noticed was that there was a lot of plastic tape to take back up later.
Food boxes are often wax coated... I always found them desirable for getting a fire going when the wood was damp...
Soap is said to promote plant growth, causing algae blooms when it washes into streams...
How it affects the compost, I couldn't say...
